Abstract
Euler’s formula for a polyhedron with V vertices, F faces and E edges and genus g states that V + F – E = 2(1 – g).
Cayley formulated a similar expression for the four Kepler-Poinsot polyhedra: bV + aF – E = 2c, using the densities
a, b and c of respectively the faces, the vertices and the polyhedron itself. We conjecture both should be united as
bV + aF – E = 2c(1 – g), or, more generally, for Archimedean polyhedra with Vj vertices and Fi faces of a given
type and densities bj and ai respectively, as bjVj + aiFi – E = 2c(1 – g). We consider examples that support our
case, but a more important aspect of this paper is perhaps the suggestion of considering higher genus Kepler-
Poinsot polyhedra. Some are unusual cases inspired by S. Gott’s pseudo-polyhedra with adjacent faces in the same
plane, while other types of these star polyhedra may be attractive from an artistic point of view. Escher was
interested in star polyhedra and in infinite tessellations, and the polyhedra proposed in the present paper combine
both topics.
